Question: Determine how many solutions exist for the system of equations. ${-2x+y = 5}$ ${-12x-2y = 20}$
Solution: Convert both equations to slope-intercept form: ${-2x+y = 5}$ $-2x{+2x} + y = 5{+2x}$ $y = 5+2x$ ${y = 2x+5}$ ${-12x-2y = 20}$ $-12x{+12x} - 2y = 20{+12x}$ $-2y = 20+12x$ $y = -10-6x$ ${y = -6x-10}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 2x+5}$ ${y = -6x-10}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.